# 1993

《紐約嫻情》是香港女歌手陳慧嫻，於 1993 年拍攝的音樂特輯；監製金廣誠，編導盧冰心；香港時間 1993 年 10 月 10 日（星期日）晚上 8 時至 9 時 05 分，於香港無綫電視翡翠台首播。

《千千闋歌》
《傻女》
《花店》
《Joe le Taxi》
《飄雪》
《碎花》
《人生何處不相逢》
《Jealousy》

《紅茶館》
《幾時再見?!》

— 文字在創用 CC 姓名標示-相同方式分享 3.0 協議之條款下提供，附加條款亦可能應用。

— 維基百科

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When this music special premiered in 1993, my TV was showing it in the background while I was having my dinner. I was not watching it. So I did not know the content. But still, I remembered the atmosphere broadcast by it.

In an afternoon in 1996, it was broadcast again. I took a VCR cassette at once to record it. However, I was already a few minutes late, missing the first part of the show.

I watched this music special after every last exam.

In 200x, I bought a TV card just to transfer the music special from the VCR to my computer. However, I have already lost both the VCR cassette and the computer file.

— Me@2022-01-26 09:30:53 PM

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2022.01.26 Wednesday ACHK

# 2.10 Extra dimension and statistical mechanics

A First Course in String Theory

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Write a double sum that represents the statistical mechanics partition function $\displaystyle{Z(a, R)}$ for the quantum mechanical system considered in Section 2.10. Note that $\displaystyle{Z(a, R)}$ factors as $\displaystyle{Z(a, R) = Z(a) \tilde{Z}(R)}$.

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Eq. (2.118):

\displaystyle{ \begin{aligned} - \frac{\hbar^2}{2m} \frac{1}{\psi(x)} \frac{d^2 \psi(x)}{dx^2} - \frac{\hbar^2}{2m} \frac{1}{\phi(x)} \frac{d^2 \phi(x)}{dx^2} &= E \\ \end{aligned}}

\displaystyle{ \begin{aligned} \psi_{k, l} (x,y) &= \psi_k (x) \phi_l (y) \\ \end{aligned}}

Eq. (2.119):

\displaystyle{ \begin{aligned} \psi_k (x) &= c_k \sin \left( \frac{k \pi x}{a} \right) \\ \end{aligned}}

\displaystyle{ \begin{aligned} \frac{d^2 \psi_k (x)}{dx^2} &= - \left( \frac{k \pi}{a} \right)^2 \psi_k (x) \\ \end{aligned}}

\displaystyle{ \begin{aligned} - \frac{\hbar^2}{2m} \frac{1}{\psi(x)} \frac{d^2 \psi(x)}{dx^2} - \frac{\hbar^2}{2m} \frac{1}{\phi(x)} \frac{d^2 \phi(x)}{dx^2} &= E \\ \frac{\hbar^2}{2m} \left( \frac{k \pi}{a} \right)^2 + \frac{\hbar^2}{2m} \left( \frac{l}{R} \right)^2 &= E \\ \end{aligned}}

\displaystyle{ \begin{aligned} E &= \frac{\hbar^2}{2m} \left[ \left( \frac{k \pi}{a} \right)^2 + \left( \frac{l}{R} \right)^2 \right] \\ \end{aligned}}

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[guess]

Index $\displaystyle{k = 1, 2, \dotsb}$ but not negative integers because, for example, $k = 1$ and $k=-1$ give physically identical states

\displaystyle{ \begin{aligned} \psi_{-1} (x) &= c_{-1} \sin \left( \frac{- \pi x}{a} \right) \\ \end{aligned}} and \displaystyle{ \begin{aligned} \psi_{1} (x) &= c_{1} \sin \left( \frac{\pi x}{a} \right) \\ \end{aligned}}.

The two wave functions give the same probability density distribution, if $c_{-1} = c_{1}$.

However, that is not the case for

\displaystyle{ \begin{aligned} \phi_l(y) &= a_l \sin \left(\frac{ly}{R}\right) + b_l \cos \left(\frac{ly}{R} \right) \\ \end{aligned}}.

So $l$ should have also negative integers as possible values: $l = \dotsb, -2, -1, 0, 1, 2, \dotsb$.

[guess]

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Eq. (2.120):

\displaystyle{ \begin{aligned} \phi_l(y) &= a_l \sin \left(\frac{ly}{R}\right) + b_l \cos \left(\frac{ly}{R} \right) \\ \end{aligned}}

Eq. (2.121):

\displaystyle{ \begin{aligned} \phi_l(y) &= \phi_l(y+2\pi R) \\ \end{aligned}}

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\displaystyle{ \begin{aligned} Z &= \sum_{k} \sum_{l} e^{- \beta E_{k,l}} \\ &= \sum_{k} \sum_{l} \exp\left\{- \beta \left( \frac{\hbar^2}{2m} \right) \left[ \left(\frac{k \pi}{a} \right)^2 + \left(\frac{l}{R}\right)^2 \right]\right\} \\ &= Z(a) \tilde{Z}(R) \\ \end{aligned}}

\displaystyle{ \begin{aligned} Z(a) &= \sum_{k=1}^\infty \exp\left[- \beta \left( \frac{\hbar^2}{2m} \right) \left(\frac{k \pi}{a} \right)^2 \right] \\ \tilde Z (R) &= \sum_{l=-\infty}^\infty \exp\left[- \beta \left( \frac{\hbar^2}{2m} \right) \left(\frac{l}{R}\right)^2 \right] \\ &= \sum_{l=-\infty}^{-1} \left( \dotsb \right) + \sum_{l=0} \left( \dotsb \right) + \sum_{l=1}^\infty \left( \dotsb \right) \\ &= 1 + 2 Z(R \pi) \\ \end{aligned}}

— Me@2022-01-19 08:45:05 PM

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